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Article

Bigolin, Francesco
Regularity results for a class of obstacle problems in Heisenberg groups. (English). Applications of Mathematics, vol. 58 (2013), issue 5, pp. 531-554
MSC: 35D30, 35J20 | MR 3104617 | Zbl 06282095 | DOI: 10.1007/s10492-013-0027-1
Keywords:
obstacle problem; weak solution; regularity; Heisenberg group
Summary:
We study regularity results for solutions $u\in H W^{1,p}(\Omega )$ to the obstacle problem $$ \int _{\Omega } \mathcal {A}(x, \nabla _{\mathbb H} u)\nabla _{\mathbb H}(v-u) {\rm d} x \geq 0 \quad \forall v\in \mathcal K_{\psi ,u}(\Omega ) $$ such that $u\geq \psi $ a.e. in $\Omega $, where $\mathcal K_{\psi ,u}(\Omega )= \{v\in HW^{1,p}(\Omega )\colon v-u\in HW_{0}^{1,p}(\Omega ) v\geq \psi \text {\rm a.e. in} \Omega \}$, in Heisenberg groups $\mathbb H^n$. In particular, we obtain weak differentiability in the $T$-direction and horizontal estimates of Calderon-Zygmund type, i.e. $$ \begin{aligned}d T\psi \in HW^{1,p}_{\rm loc}(\Omega )&\Rightarrow Tu\in L^p_{\rm loc}(\Omega ), |\nabla _{\mathbb H}\psi |^p\in L^{q}_{\rm loc}(\Omega )&\Rightarrow |\nabla _{\mathbb H} u|^p \in L^q_{\rm loc}(\Omega ), \end{aligned}d $$ where $2<p<4$, $q>1$.
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